Priyanka and Ethan were asked to find an explicit formula for the sequence $-3,-14,-25,-36,...$, where the first term should be $g(1)$. Priyanka said the formula is $g(n)=-3-11n$. Ethan said the formula is $g(n)=-3+11n$. Which one of them is right? Choose 1 answer: Choose 1 answer: (Choice A) A Only Priyanka (Choice B) B Only Ethan (Choice C) C Both Priyanka and Ethan (Choice D) D Neither Priyanka nor Ethan
Answer: The general explicit formula for arithmetic sequences is ${a_1}+{d}(n-1)$, where ${a_1}$ is the first term and $ d$ is the common difference. The first term is ${-3}$ and the common difference is ${-11}$. ${-11\,\curvearrowright}$ ${-11\,\curvearrowright}$ ${-11\,\curvearrowright}$ ${-3},$ $-14,$ $-25,$ $-36,...$ We get the following formula. $g(n)={-3}{-11}(n-1)$ We can now see that $g(n)=-3-11n$ is not a correct formula, because the constant difference is added one extra time for each term. For instance, according to this formula, the value of the first term would be: $g(1)=-3-11\cdot1=-14$. However, according to our table of values, $g(1)=-3$. So Priyanka is definitely wrong. What about Ethan? We can see that $g(n)=-3+11n$ is also not a correct formula, because the constant difference according to this formula is $11$, while the actual constant difference is $-11$. So Ethan is also wrong. Neither Priyanka nor Ethan got a correct explicit formula.